Yes, why not?
I'm going to crib from The Crystal Spheres, Ask a Foolish Question and The Naked God. And The Last Question for good measure.
You say we have a K-IV civilization - something like the originator race of AAFQ:
Of the race that built him, the less said the better. They also Knew,
and never said whether they found the knowledge pleasant. They built
Answerer as a service to less-sophisticated races, and departed in a
unique manner. Where they went, only Answerer knows. Because Answerer
So, this K-IV race has godlike powers, as a K-IV race is wont to have. They decide to help somewhat the less-sophisticated races; but you cannot give matches to children, and FTL travel technology is potentially arming those children with RPGs.
The K-IV guys want to help the more backward races, but they don't want one of them, say, to enslave all the others.
And it would happen because, to have FTL, you need to have, say, adeledicnander generators, and if you have those, it would be trivial to reconfigure them into weapons of mass destruction.
So they cheat. They design and build machines that do something simple, much like a dynamo does. But while doing the only thing they actually can do, those machines also generate, as a secondary effect, specific spurious signals - just like the EM interference of a real dynamo. There is no reason they shouldn't, and there's no reason to investigate those - everybody knows they're just interferences.
Very far away, and yet not very far at all, in a dimension someone might call hyperspace, other machines lurk, and detect those interferences. When they establish that a viable "FTL engine" has been activated, they activate the real generators and pluck the engine and whatever it is attached to out of normal space, to reintegrate it in the appropriate location of the space-time continuum. This is
what the Naked God does in the third book of The Reality Dysfunction series: it can project massive wormholes on a galactic scale.
At any moment, at the K-IV people's whim, the FTL "engines" could stop working, or stop working for someone specific, or malfunction in any way. Because they're not FTL engines at all, they're just bells. You ring the bell, and if the correct note is struck, the bellboy comes and the service gets done - but it's not by any inherent power of the bell.
Being comparatively simple, it would be possible to replicate these "FTL engines" with little trouble, and not being FTL engines at all, they can't be reengineered or understood.
Actually, they could be just dynamos. A heavy dynamo with an alternated osmium-aluminum rotor connected to specially shaped coils. When operating, the device releases extremely weak gravitational waves, and inside it there are characteristic electromagnetic waves of exactly the same frequency. The source of both disturbances is pinpointed by the hyperspace controllers - using something like Iain M. Banks' Culture's effectors - that are able to analyze the device and "read" the current in the coils, then translate this information into jump parameters (say, every ampère of current in the coils is one light-year in the same direction as the dynamo axis).
Using the same trick, the hyperspace controllers can inject a current in a properly shaped coil, and "leak" information about e.g. nearby masses or FTL engine "wakes" or things like that.
So you have it - from the lesser races' point of view, a not too complex contraption made of rare earths magnets and both very dense and very light metals, rotating at a specific speed, opens a "portal" - then the current in the coils "drives" the ship through the portal at a "pseudo-speed" that depends on the ship's mass, the current in the coils, their diameter, the voltage, whether they're immersed in a magnetic field and so on.
Lots of experimentation and more and more outlandish theories would ensue, but the operating principle of the machines would remain mysterious.
This does not mean that experimentation is useless; it leads to the discovery of the FTL "rules". Which can be as arbitrary as we need.
Rotate them slower than the threshold, and nothing at all happens. Increase the current or the coil area, and the distance covered changes - but that, while almost making sense, is of very little practical use. The engines might consume so little that they can reach any distance; but, after a jump that is never longer than, say, ten light-years, nothing can jump to or from the same volume of space (say, ten light-minute in radius) for some time, say four hours. Attempting to jump in a "depleted" or "hot" volume before it has "recovered" or "cooled down" means destruction of the engine (or maybe the ship?). This means that information speed, using relay couriers, is now one light-year per minute; ship speed is 2.5 light-years per hour. Also, a volume of space can be made impervious to FTL by having several FTL drones, ten light-minutes apart, hop around the whole volume. Precise knowledge of the drone schedule allows FTL travel in and out of the volume; jumping at random means almost surely smashing the engine into the "depletion shield".